?For the following exercises, use the information provided to solve the problem.Let

For the following exercises, use the information provided to solve the problem. Let \(w(x,\ y,\ z)=xy\cos z\), where \(x=t,\ y=t^2\), and \(z=\arcsin t\). Find \(\frac{d w}{d t}\). Text Transcription: w(x,\ y,\ z)=xy\cos z x=t,\ y=t^2 z=\arcsin \frac{d w}{d t}

For the following exercises, use the information provided to solve the problem. Let \(w(t, v)=e^{t v}\) where \(t\) = \(r\) + \(s\) and \(v\) = \(rs\). Find \(\frac{\partial w}{\partial r} \text { and } \frac{\partial w}{\partial s}\) Text Transcription: w(t, v)=e^{t v} t = r + s and v = rs \frac{\partial w}{\partial r} and \frac{\partial w}{\partial s}

For the following exercises, use the information provided to solve the problem. If \(w=5 x^{2}+2 y^{2}, x=-3 s+t\), and \(y=s-4 t\), find \(\frac{\partial w}{\partial s} \text { and } \frac{\partial w}{\partial t}\). Text Transcription: w=5 x^{2}+2 y^{2}, x=-3 s+t y=s-4 t \frac{\partial w}{\partial s} \text { and } \frac{\partial w}{\partial t}

For the following exercises, use the information provided to solve the problem. Let \(w=x y^{2}, x=5 \cos (2 t)\), and \(y=5 \sin (2 t)\), find \(\frac{\partial w}{\partial t}\). Text Transcription: w=x y^{2}, x=5 \cos (2 t) y=5 \sin (2 t) \frac{\partial w}{\partial t}

For the following exercises, use the information provided to solve the problem. If \(f(x,\ y)=xy,\ x=r\cos\theta, and \(y=r \sin \theta\), find \(\frac{\partial f}{\partial r}\) and express the answer in terms of \(r\) and \(\theta\). Text Transcription: f(x,\ y)=xy,\ x=r\cos\theta, and \(y=r \sin \theta \frac{\partial f}{\partial r} r theta

For the following exercises, use the information provided to solve the problem. Suppose \(f(x,\ y)=x+y,u=e^x\sin y,\ x=t^2\), and \(y=\pi t\), where \(x=r \cos \theta \text { and } y=r \sin \theta\). Find \(\frac{\partial f}{\partial \theta}\). Text Transcription: f(x,\ y)=x+y,u=e^x\sin y,\ x=t^2 y=\pi t x=r \cos \theta y=r \sin \theta \frac{\partial f}{\partial \theta}

For the following exercises, find \(\frac{d f}{d t}\) using the chain rule and direct substitution. \(f(x,\ y)=x^2+y^2,\ \quad x=t,\ y=t^2\) Text Transcription: f(x,\ y)=x^2+y^2,\ \quad x=t,\ y=t^2

For the following exercises, find \(\frac{d f}{d t}\) using the chain rule and direct substitution. \(f(x,\ y)=\sqrt{x^2+y^2},\ y=t^2,\ x=t\) Text Transcription: f(x,\ y)=\sqrt{x^2+y^2},\ y=t^2,\ x=t

For the following exercises, find \(\frac{d f}{d t}\) using the chain rule and direct substitution. \(f(x,y)=xy,\ x=1-\sqrt{t},\ y=1+\sqrt{t}\) Text Transcription: f(x,y)=xy,\ x=1-\sqrt{t},\ y=1+\sqrt{t}

For the following exercises, find \(\frac{d f}{d t}\) using the chain rule and direct substitution. \(f(x,\ y)=\frac{x}{y},\ x=e^t,\ y=2e^t\) Text Transcription: f(x,\ y)=\frac{x}{y},\ x=e^t,\ y=2e^t

For the following exercises, find \(\frac{d f}{d t}\) using the chain rule and direct substitution. \(f(x,y)=\ln(x+y),\ x=e^t,\ y=e^t\) Text Transcription: f(x,y)=\ln(x+y),\ x=e^t,\ y=e^t

For the following exercises, find \(\frac{d f}{d t}\) using the chain rule and direct substitution. \(f(x,y)=x^4,\quad\ x=t,\ y=t\) Text Transcription: f(x,y)=x^4,\quad\ x=t,\ y=t

For the following exercises, find \(\frac{d f}{d t}\) using the chain rule and direct substitution. Let \(w(x,\ y,\ z)=x^2+y^2+z^2\), \(x=\cos t,\ y=\sin t,\quad\text{ and }\quad z=e^t\). Express \(w\) as a function of \(t\) and find \(\frac{d w}{d t}\) directly. Then, find \(\frac{d w}{d t}\) using the chain rule. Text Transcription: w(x,\ y,\ z)=x^2+y^2+z^2 x=\cos t,\ y=\sin t,\quad\text{ and }\quad z=e^t w t \frac{d w}{d t}

For the following exercises, find \(\frac{d f}{d t}\) using the chain rule and direct substitution. Let \(z=x^{2} y\), where \(x=t^{2} \text { and } y=t^{3}\). Find \(\frac{d z}{d t}\). Text Transcription: z=x^{2} y x=t^{2} y=t^{3} \frac{d z}{d t}

For the following exercises, find \(\frac{d f}{d t}\) using the chain rule and direct substitution. Let \(u=e^{x} \sin y, \quad \text { where } x=t^{2} \text { and } y=\pi t\). Find \(\frac{d u}{d t}\) when \(x=\ln 2 \text { and } y=\frac{\pi}{4}\) Text Transcription: u=e^{x} \sin y, \quad x=t^{2} y=\pi t \frac{d u}{d t} x=\ln 2 y=\frac{\pi}{4}

For the following exercises, find \(\frac{d y}{d x}\) using partial derivatives. \(\sin (6 x)+\tan (8 y)+5=0\) Text Transcription: \sin (6 x)+\tan (8 y)+5=0

For the following exercises, find \(\frac{d y}{d x}\) using partial derivatives. \(x^{3}+y^{2} x-3=0\) Text Transcription: x^{3}+y^{2} x-3=0

For the following exercises, find \(\frac{d y}{d x}\) using partial derivatives. \(\sin (x+y)+\cos (x-y)=4\) Text Transcription: \sin (x+y)+\cos (x-y)=4

For the following exercises, find \(\frac{d y}{d x}\) using partial derivatives. \(x^{2}-2 x y+y^{4}=4\) Text Transcription: x^{2}-2 x y+y^{4}=4

For the following exercises, find \(\frac{d y}{d x}\) using partial derivatives. \(x e^{y}+y e^{x}-2 x^{2} y=0\) Text Transcription: x e^{y}+y e^{x}-2 x^{2} y=0

For the following exercises, find \(\frac{d y}{d x}\) using partial derivatives. \(x^{2 / 3}+y^{2 / 3}=a^{2 / 3}\) Text Transcription: frac_dy/dx x^2/3+y^2/3=a^2/3

For the following exercises, find \(\frac{d y}{d x}\) using partial derivatives. x cos(xy) + y cos x = 2 Text Transcription: frac_dy/dx

For the following exercises, find \(\frac{d y}{d x}\) using partial derivatives. \(e^{x y}+y e^{y}=1\) Text Transcription: frac_dy/dx e^xy+ye^y=1

For the following exercises, find \(\frac{d y}{d x}\) using partial derivatives. \(x^{2} y^{3}+\cos y=0\) Text Transcription: frac_dy/dx x^2_y^3+cos_y=0

For the following exercises, find \(\frac{d y}{d x}\) using partial derivatives. Find \(\frac{d z}{d t}\) using the chain rule where \(z=3 x^{2} y^{3}\), \(x=t^{4}\) and \(y=t^{2}\). Text Transcription: frac_dy/dx frac_dz/dt z=3 x^2_y^3 x=t^4 y=t^2

For the following exercises, find \(\frac{d y}{d x}\) using partial derivatives. Let \(z = 3 cos x ? sin(xy)\), \(x=\frac{1}{t}\), and y = 3t. Find \(\frac{d z}{d t}\). Text Transcription: frac_dy/dx z=3cos.x?sin.xy x=frac_1/t frac_dz/dt

For the following exercises, find \(\frac{d y}{d x}\) using partial derivatives. Let \(z=e^{1-x y}\),\( \quad x=t^{1 / 3}\) and \(y=t^{3}\). Find \(\frac{d z}{d t}\). Text Transcription: frac_dy/dx z=e^1-xy quad.x=t^1/3 y=t^3 frac_dz/dt

For the following exercises, find \(\frac{d y}{d x}\) using partial derivatives. Find (\frac{d z}{d t}\) by the chain rule where \(z=\cosh ^{2}(x y)\),\(\quad x=\frac{1}{2} t\), and \(y=e^{t}\). Text Transcription: frac_dy/dx frac_dz/dt z=cosh^2(xy) quad.x=frac_1/2_t y=e^t

For the following exercises, find \(\frac{d y}{d x}\) using partial derivatives. Let \(z=\frac{x}{y}\), \(\quad x=2 \cos u\), and y = 3 sin v. Find \(\frac{\partial z}{\partial u}\) and \(\frac{\partial z}{\partial v}\). Text Transcription: frac_dy/dx z=frac_x/y quad_x=2_cos.u frac_partial.z/partial.u frac_partial.z/partial.v

For the following exercises, find \(\frac{d y}{d x}\) using partial derivatives. Let \z9=e^{x^{2} y}\), where \(x=\sqrt{u v}\) and \(y=\frac{1}{v}\). Find \(\frac{\partial z}{\partial u}\) and \(\frac{\partial z}{\partial v}\). Text Transcription: frac_dy/dx z=e^x^2_y x=sqrt.uv y=frac.1/v frac_partial.z/partial.u frac_partial.z/partial.v

For the following exercises, find \(\frac{d y}{d x}\) using partial derivatives. If \(z=x y e^{x / y}\), \(x=r \cos \theta\), and \(y=r \sin \theta\), find \(\frac{\partial z}{\partial r}\) and \(\frac{\partial z}{\partial \theta}\) when r = 2 and \(\theta=\frac{\pi}{6}\). Text Transcription: frac_dy/dx z=xye^x/y x=r_cos_theta frac_partial.z/partial.r frac_partial.z/partial.theta theta=frac_pi/6

For the following exercises, find \(\frac{d y}{d x}\) using partial derivatives. Find \(\frac{\partial w}{\partial s}\) if \(w=4 x+y^{2}+z^{3}\), \(x=e^{r s^{2}}\), \(y=\ln \left(\frac{r+s}{t}\right)\) and \(z=r s t^{2}\). Text Transcription: frac_dy/dx frac_partial.w/partial.s w=4x+y^2+z^3 x=e^rs^2 y=ln_left_frac_r+s/t.right) z=rst^2

For the following exercises, find \(\frac{d y}{d x}\) using partial derivatives. If \(w=\sin (x y z)\), \(x=1-3 t, y=e^{1-t}\) and z = 4t, find \(\frac{\partial w}{\partial t}\). Text Transcription: frac_dy/dx w=sin(xyz) x=1-3t,y=e^1-t frac_partial.w/partial.t

For the following exercises, use this information: A function f (x, y) is said to be homogeneous of degree n if \(f(t x, t y)=t^{n} f(x, y)\). For all homogeneous functions of degree n, the following equation is true: \(x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}=n f(x, y)\). Show that the given function is homogeneous and verify that \(x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}=n f(x, y)\). \(f(x, y)=3 x^{2}+y^{2}\) Text Transcription: f(tx,ty)=t^n_f(x,y) x_frac_partial.f/partial.x+y_frac_partial.f/partial.y}=nf(x,y) x_frac_partial.f/partial.x+y_frac_partial.f/partial.y}=nf(x,y) f(x,y)=3x^2+y^2

For the following exercises, use this information: A function f (x, y) is said to be homogeneous of degree n if \(f(t x, t y)=t^{n} f(x, y)\). For all homogeneous functions of degree n, the following equation is true: \(x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}=n f(x, y)\). Show that the given function is homogeneous and verify that \(x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}=n f(x, y)\). \(f(x, y)=\sqrt{x^{2}+y^{2}}\) Text Transcription: f(tx,ty)=t^n_f(x,y) x_frac_partial.f/partial.x+y_frac_partial.f/partial.y}=nf(x,y) x_frac_partial.f/partial.x+y_frac_partial.f/partial.y}=nf(x,y) f(x,y)=sqrt_x^2+y^2

For the following exercises, use this information: A function f (x, y) is said to be homogeneous of degree n if \(f(t x, t y)=t^{n} f(x, y)\). For all homogeneous functions of degree n, the following equation is true: \(x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}=n f(x, y)\). Show that the given function is homogeneous and verify that \(x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}=n f(x, y)\). \(f(x, y)=x^{2} y-2 y^{3}\) Text Transcription: f(tx,ty)=t^n_f(x,y) x_frac_partial.f/partial.x+y_frac_partial.f/partial.y}=nf(x,y) x_frac_partial.f/partial.x+y_frac_partial.f/partial.y}=nf(x,y) f(x,y)=x^2y-2y^3

For the following exercises, use this information: A function f(x, y) is said to be homogeneous of degree n if \(f(t x, t y)=t^{n} f(x, y)\). For all homogeneous functions of degree n, the following equation is true: \(x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}=n f(x, y)\). Show that the given function is homogeneous and verify that \(x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}=n f(x, y)\). The volume of a right circular cylinder is given by \(V(x, y)=\pi x^{2} y\), where x is the radius of the cylinder and y is the cylinder height. Suppose x and y are functions of t given by \(x=\frac{1}{2} t\) and \(y=\frac{1}{3} t\) so that x and y are both increasing with time. How fast is the volume increasing when x = 2 and y = 5? Text Transcription: f(tx,ty)=t^n_f(x,y) x_frac_partial.f/partial.x+y_frac_partial.f/partial.y}=nf(x,y) x_frac_partial.f/partial.x+y_frac_partial.f/partial.y}=nf(x,y) V(x,y)=pi.x^2y x=frac_1/2t y=frac_1/3

For the following exercises, use this information: A function f (x, y) is said to be homogeneous of degree n if \(f(t x, t y)=t^{n} f(x, y)\). For all homogeneous functions of degree n, the following equation is true: \(x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}=n f(x, y)\). Show that the given function is homogeneous and verify that \(x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}=n f(x, y)\). The pressure P of a gas is related to the volume and temperature by the formula PV = kT, where temperature is expressed in kelvins. Express the pressure of the gas as a function of both V and T. Find \(\frac{d P}{d t}\) when \(k = 1, \frac{d V}{d t}=2 \mathrm{~cm}^{3} / \mathrm{min}, \frac{d T}{d t}=\frac{1}{2} \mathrm{~K} / \mathrm{min}, \quad V=20 \mathrm{~cm}^{3}\) and T = 20°F. Text Transcription: f(tx,ty)=t^n_f(x,y) x_frac_partial.f/partial.x+y_frac_partial.f/partial.y}=nf(x,y) x_frac_partial.f/partial.x+y_frac_partial.f/partial.y}=nf(x,y) frac_dP/dt k=1,frac_dV/dt=2_mathrm.~cm^3/mathrm.min,frac_dT/dt}=frac_1/2.mathrm~K.mathrm.min,quadV=20mathrm~cm^3

For the following exercises, use this information: A function f (x, y) is said to be homogeneous of degree n if \(f(t x, t y)=t^{n} f(x, y)\). For all homogeneous functions of degree n, the following equation is true: \(x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}=n f(x, y)\). Show that the given function is homogeneous and verify that \(x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}=n f(x, y)\). The radius of a right circular cone is increasing at 3 cm/min whereas the height of the cone is decreasing at 2 cm/min. Find the rate of change of the volume of the cone when the radius is 13 cm and the height is 18 cm. Text Transcription: f(tx,ty)=t^n_f(x,y) x_frac_partial.f/partial.x+y_frac_partial.f/partial.y}=nf(x,y) x_frac_partial.f/partial.x+y_frac_partial.f/partial.y}=nf(x,y)

For the following exercises, use this information: A function f (x, y) is said to be homogeneous of degree n if \(f(t x, t y)=t^{n} f(x, y)\). For all homogeneous functions of degree n, the following equation is true: \(x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}=n f(x, y)\). Show that the given function is homogeneous and verify that \(x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}=n f(x, y)\). The volume of a frustum of a cone is given by the formula, \(V=\frac{1}{3} \pi z\left(x^{2}+y^{2}+x y\right)\), where x is the radius of the smaller circle, y is the radius of the larger circle, and z is the height of the frustum (see figure). Find the rate of change of the volume of this frustum when x = 10 in., y = 12 in., and z = 18 in. Text Transcription: f(tx,ty)=t^n_f(x,y) x_frac_partial.f/partial.x+y_frac_partial.f/partial.y}=nf(x,y) x_frac_partial.f/partial.x+y_frac_partial.f/partial.y}=nf(x,y) V=frac1/3pi.z.left(x^2+y^2+xy.right)

A closed box is in the shape of a rectangular solid with dimensions x, y, and z. (Dimensions are in inches.) Suppose each dimension is changing at the rate of 0.5 in./min. Find the rate of change of the total surface area of the box when x = 2 in., y = 3 in., and z = 1 in.

The total resistance in a circuit that has three individual resistances represented by x, y, and z is given by the formula \(R(x, y, z)=\frac{x y z}{y z+x z+x y}\) . Suppose at a given time the x resistance is \(100 \Omega\), the y resistance is \(200 \Omega\), and the z resistance is \(300 \Omega\). Also, suppose the x resistance is changing at a rate of \(2 \Omega / \mathrm{min}\), the y resistance is changing at the rate of \(1 \Omega / \mathrm{min}\), and the z resistance has no change. Find the rate of change of the total resistance in this circuit at this time. Text Transcription: R(x, y, z) = xyz / yz + xz + xy 100 Omega 200 Omega 300 Omega 2 Omega / min 1 Omega / min

The temperature T at a point (x, y) is T(x, y) and is measured using the Celsius scale. A fly crawls so that its position after t seconds is given by \(x=\sqrt{1+t}\) and \(y=2+\frac{1}{3} t\), where x and t are measured in centimeters. The temperature function satisfies \(T_{x}(2,3)=4\) and \(T_{y}(2,3)=3\). How fast is the temperature increasing on the fly’s path after 3 sec? Text Transcription: x = sqrt 1+t y = 2 + 1/3 t T_x (2,3) = 4 T_y (2,3) = 3

The x and y components of a fluid moving in two dimensions are given by the following functions: u(x, y) = 2y and v(x, y) = ?2x; \(x \geq 0 ; y \geq 0\). The speed of the fluid at the point (x, y) is \(s(x, y)=\sqrt{u(x, y)^{2}+v(x, y)^{2}}\) . Find \(\frac{\partial s}{\partial x}\) and \(\frac{\partial s}{\partial y}\) using the chain rule. Text Transcription: x geq 0 ; y geq 0 s(x, y) = sqrt u(x, y)^2 + v(x, y)^2 partial s / partial x partial s / partial y

Let u = u(x, y, z), where x = x(w, t), y = y(w, t), z = z(w, t), w = w(r, s), and t = t(r, s). Use a tree diagram and the chain rule to find an expression for \(\frac{\partial u}{\partial r}\) . Text Transcription: partial u / partial r